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In the process of writing an article on curvature we got entirely distracted by making a geogebra worksheet showing the tangent, normal and osculating circle to any smooth function. The meaning and mathematics of all these terms is revealed in this article, but if you fancy getting your hands dirty yourself, have a play with the worksheet below. Please post your favourite function as a comment – the curvier and wigglier the better!

You can use this geogebra worksheet to see the tangent, normal and osculating circle of any smooth curve you choose - just change the equation f(x) in the left-hand panel.

These lucky people are climbing around a 22 feet (6.7 metres) tall structure composed of 384 softball bats, 130 soft balls and a couple of thousand pounds of steel. The structure represents a Sierpinski tetrahedron: a fractal which has finite volume but infinite area. The image only shows an approximation of the fractal of course, as it would be impossible to make a full-on Sierpinski tetrahedon with its infinite intricacy, but it's beautiful anyway!

IMage courtesy of the Department of Civil and Environmental Engineering, Imperial College London.

This is one of friend of PlusAhmer Wadee's favourites images from the book 50 visions of mathematics. It is of a demonstration at Imperial College in 1887 of the mathematical principles behind (or should that be underneath?) the Forth Bridge. The bridge was the largest spanning bridge in the world at the time and the technique behind it was an innovation, essentially balancing the forces involved using cantilevers. The
men on the chairs (Sir John Fowler and Benjamin Baker) represent the piers of the bridge and the load on the bridge, in this case Kaichi Watanabe, one of the first Japanese engineers to study in the UK, is supported by the tension (in the men's arms and in the ropes to the anchors) and compression in the structure.

If you love physics and making movies then this is for you. The Foundational Questions Institute (FQXi) is excited to present Show me the physics!, its first-ever video contest. Anyone can submit a video conveying the joys of physics to win a top prize of $10,000, and there are very attractive runner-up prizes too.

Whether you're a physicist or just a physics geek, and whether it's the geometry of space time (see top video on the right) or quantum immortality (see bottom video), all sorts of submissions are welcome. The aim is to enthuse non-physicists and provide a creative and visual space for the discussion and exchange of ideas. Examples of suitable topics are:

Unsolved physics mysteries

Physics experiments being carried out

Tales of physics discoveries

Accounts of how physics has improved our lives

Physicists, inventors, teachers, and others talking about their passion for physics

Fictional stories in which real physics plays a central role.

The closing date is August 8, 2014. See here here for rules and submission guidelines and here to see the current entries.

To mark Germany's historic win over Brazil in the World Cup semifinal this week, Lasse Rempe-Gillen (from the University of Liverpool) created this beautiful image. It shows the behaviour of a model that describes the phenomenon of phase-locking, something that can be seen in the synchronising flashes of fireflies or when a roaring stadium of football supporters gradually clap or stamp in unison. The image is related to recent research and you can read more in our news story Maths, metronomes and fireflies.

The grey parts of the image show where the model behaves chaotically – here even small changes in where you start can cause drastically different results in the model. The coloured parts of the image show where the model behaves in a more regular fashion where small differences won't dramatically change the results. This is because the model has attractors, special sets of conditions that create similar behaviour, either settling on a single outcome (called a fixed point) or running through a predictable cycle of outcomes. And in honour of the historic 7-1 score from the match, Rempe-Gillen's image has attractors of period 7 (with a repeating cycle of 7 points) and period 1 (a fixed point).

In contrast his image below has no periodic attractors, symbolising the other, goalless, semifinal between Argentina and Holland.